Geodesic
A comparison result related to Gauss measure
[Un résultat de comparaison relatif à la mesure de Gauss]
Betta, M.Francesca1 ; Brock, Friedman2 ; Mercaldo, Anna3 ; Posteraro, M.Rosaria3
1 Dipartimento di Matematica, Seconda Università di Napoli, via Vivaldi 43, 81100 Caserta, Italy
2 Department of Mathematics, University of Missouri-Columbia, Columbia, MO 65211, USA
3 Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy
Comptes Rendus. Mathématique, Tome 334 (2002) no. 6, pp. 451-456.

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In this paper we prove a comparison result for weak solutions to linear elliptic problems of the type

-(a ij (x)u x i ) x j =f(x)ϕ(x) in Ω,u=0 on ∂Ω,
where Ω is an open set of ℝ n (n⩾2), ϕ(x)=(2π)−n/2exp(−|x|2/2), aij(x) are measurable functions such that aij(x)ξiξj⩾ϕ(x)|ξ|2 a.e. x∈Ω, ξ∈ℝ n and f(x) is a measurable function taken in order to guarantee the existence of a solution u∈H 0 1 (ϕ,Ω) of (1.1). We use the notion of rearrangement related to Gauss measure to compare u(x) with the solution of a problem of the same type, whose data are defined in a half-space and depend only on one variable.

Dans cette note on démontre un résultat de comparaison pour les solutions faibles de problèmes elliptiques linéaires du type

-(a ij (x)u x i ) x j =f(x)ϕ(x) dans Ω,u=0 sur ∂Ω,
où Ω est un ouvert de ℝ n (n⩾2), ϕ(x)=(2π)−n/2exp(−|x|2/2), aij(x) sont des fonctions mesurables telles que aij(x)ξiξj⩾ϕ(x)|ξ|2 p.p. x∈Ω, ξ∈ℝ n et f(x) est une fonction mesurable telle qu'il existe une solution u de (0.1), dans H 0 1 (ϕ,Ω). On utilise la notion de rearrangement relatif à la mesure de Gauss pour comparer u(x) avec la solution d'un problème du même type, dont les données sont définies dans un demi plan et dépendent d'une variable seulement.

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DOI : 10.1016/S1631-073X(02)02295-1

Betta, M.Francesca 1 ; Brock, Friedman 2 ; Mercaldo, Anna 3 ; Posteraro, M.Rosaria 3

1 Dipartimento di Matematica, Seconda Università di Napoli, via Vivaldi 43, 81100 Caserta, Italy
2 Department of Mathematics, University of Missouri-Columbia, Columbia, MO 65211, USA
3 Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy 
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     title = {A comparison result related to {Gauss} measure},
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Betta, M.Francesca; Brock, Friedman; Mercaldo, Anna; Posteraro, M.Rosaria. A comparison result related to Gauss measure. Comptes Rendus. Mathématique, Tome 334 (2002) no. 6, pp. 451-456. doi : 10.1016/S1631-073X(02)02295-1. http://geodesic.mathdoc.fr/articles/10.1016/S1631-073X(02)02295-1/

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